This post is an expanded version of this MSE post.
Assume that $(M, \omega)$ is a symplectic manifold which is equiped with a Riemannian metric.
Is there a symplectic structure $\omega '$ which is a harmonic $2$-form? Can one choose such a $\omega'$ such that it would be de Rham cohomolgue to the initial form $\omega$?
We consider the following particular case:
For a Riemannian manifold $(M,g)$, we equip the tangent bundle $TM$ with the Sasaki metric $g_s$ and the natural symplectic structure $\omega $ arising from the canonical structure on the cotangent bundle.
Under which conditions on the Riemannian manifold $(M,g)$, is the symplectic $2$-form $\omega$ a harmonic form on $(TM,g_s)$?